Eyes not following body
There is something wrong with the parenting: I solved the problem selecting the L eye, shift selecting the L eye rig, and pressing Ctrl P choosing "Armature deform" option (This option doesn't affect the weights you've already set). Do the same with R eye.

Then, personally, I would prefere to merge all rigs into one (Ctrl J, but you will have to rename some bones, to prevent name conflicts).